COMMUN. MATH. SCI. c  2011 International Press Vol. 9, No. 4, pp. 1033–1050 ON THE STRONG SOLUTION OF A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS THAT ARISE IN THE PRICING OF MORTGAGE BACKED SECURITIES ∗ RANA D. PARSHAD † , DERVIS BAYAZIT ‡ , NATHANIEL S. BARLOW § , AND V. RAMCHANDRA PRASAD ¶ Abstract. We consider a reduced form pricing model for mortgage backed securities, formulated as a non-linear partial differential equation. We prove that the model possesses a weak solution. We then show that under additional regularity assumptions on the initial data, we also have a mild solution. This mild solution is shown to be a strong solution via further regularity arguments. We also numerically solve the reduced model via a Fourier spectral method. Lastly, we compare our numerical solution to real market data. We observe interestingly that the reduced model captures a number of recent market trends in this data, that have escaped previous models. Key words. Mortgage backed security, reduced modeling, mild solution, strong solution, Fourier spectral method. AMS subject classifications. 35A01, 35D35, 76M22, 91G20, 91G80. 1. Introduction Real world modelling often involves the description of complex and non linear processes, which can be difficult to deal with in their most general forms. One school of thought considers various limiting cases of a parameter or variable of interest in a model, that might lead to a simplification. The equations obtained via these lim- its, albeit unrealistic, might be easier to analyse or perform numerical computations on. This is particularly favourable in specific scenarios, where there is evidence that actually taking these limits is feasible. Recently this approach has been carried out successfully in analysing partial differential equations describing fluid convection [17], and in analysing partial differential equations describing fluid convection in a porous media [11]. In [11] we proved convergence of global attractors and stationary statisti- cal properties of the Darcy-Boussinesq system to those of the infinite Darcy-Prandtl number model. This model is a reduced variant of the Darcy-Boussinesq system when the permeability (which is a measure of a porous medium to transmit a fluid) approaches 0. In tightly packed media such as granite, limestone or dolomite, taking this limit is feasible [20]. This makes a strong case for the use of the infinite Darcy- Prandtl number model, in applications like hydraulic fracturing or thermal induced oil recovery, in particular if the drilling scenarios are in very tightly packed media [20]. In [12] we started a program where we adopted some of the methodology of the above mentioned works to study mortgage backed securities. Recall that a mortgage backed security is formed by pooling together a group of mortgages and then selling this pool as a security to investors. These constitute over a trillion dollar issuance in * Received: November 4, 2010; accepted (in revised version): March 19, 2011. Communicated by David Cai. † King Abdullah University of Science and Technology, Applied Mathematics and Computational Science, Thuwal 23955-6900, Kingdom of Saudi Arabia (rana.parshad@kaust.edu.sa). ‡ Federal Home Loan Bank of Atlanta, 1475 Peachtree Street, N.E. Atlanta, GA 30309, USA. § Center for Computational Research, State Universty of New York at Buffalo, Buffalo, NY 14203, USA. ¶ Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle 517325, Andhra Pradesh, India. 1033